Thank you for visiting A variable Z is normally distributed with µ 54 and σ 12 3 Find the following probabilities a P 40 Z 55 7 b P. This page is designed to guide you through key points and clear explanations related to the topic at hand. We aim to make your learning experience smooth, insightful, and informative. Dive in and discover the answers you're looking for!
Answer :
To solve these probability problems, we'll use the properties of the normal distribution. A normal distribution is defined by its mean (µ) and standard deviation (σ). For the given variable Z, we have a mean (µ) of 54 and a standard deviation (σ) of 12.3.
To find probabilities for a normally distributed variable, we typically use the standard normal distribution (a normal distribution with a mean of 0 and a standard deviation of 1) and Z-scores. The Z-score formula is:
[tex]Z = \frac{(X - µ)}{σ}[/tex]
Here’s how to approach each part:
a. P(40 < Z \leq 55.7):
Find the Z-score for X = 40:
[tex]Z_{40} = \frac{(40 - 54)}{12.3} = -1.14[/tex]Find the Z-score for X = 55.7:
[tex]Z_{55.7} = \frac{(55.7 - 54)}{12.3} = 0.14[/tex]Use standard normal distribution tables or calculators to find the probabilities:
- P(Z < 55.7) corresponds to the cumulative probability for Z = 0.14.
- P(Z < 40) corresponds to the cumulative probability for Z = -1.14.
Calculate the probability:
- P(40 < Z \leq 55.7) = P(Z < 55.7) - P(Z < 40)
b. P(Z = 64.9):
In a continuous probability distribution like the normal distribution, the probability of a single point is 0. Therefore,
- P(Z = 64.9) = 0.
c. P(Z > 54):
Since 54 is the mean, we know the normal distribution is symmetric around the mean.
- P(Z > 54) = 0.5 because half of the values in a normal distribution fall above the mean.
d. P(Z < 48.1):
Calculate the Z-score for X = 48.1:
[tex]Z_{48.1} = \frac{(48.1 - 54)}{12.3} = -0.48[/tex]Find P(Z < -0.48) using a standard normal distribution table or calculator.
e. P(Z ≠ 63.4):
The probability of Z not being a specific value in a continuous distribution is always 1. Therefore,
- P(Z ≠ 63.4) = 1.
f. P(Z ≤ 70):
Calculate the Z-score for X = 70:
[tex]Z_{70} = \frac{(70 - 54)}{12.3} = 1.30[/tex]Find P(Z ≤ 1.30) using a standard normal distribution table or calculator.
g. P(Z < 38.2 OR Z > 57.3):
Calculate the Z-score for X = 38.2:
[tex]Z_{38.2} = \frac{(38.2 - 54)}{12.3} = -1.28[/tex]Calculate the Z-score for X = 57.3:
[tex]Z_{57.3} = \frac{(57.3 - 54)}{12.3} = 0.27[/tex]Find the cumulative probabilities for these Z-scores:
- P(Z < 38.2) = P(Z < -1.28)
- P(Z > 57.3) = 1 - P(Z < 57.3) = 1 - P(Z < 0.27)
Combine for total probability:
- P(Z < 38.2 OR Z > 57.3) = P(Z < 38.2) + P(Z > 57.3)
Use standard normal distribution tables or calculators for these calculations to get the actual probability values.
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